11th World Congress on Computational Mechanics (WCCM XI)
5th European Conference on Computational Mechanics (ECCM V)
6th European Conference on Computational Fluid Dynamics (ECFD VI)
E. Oñate, J. Oliver and A. Huerta (Eds)
STRUCTURAL HEALTH MONITORING OF STAY CABLES BY THE
SCRUTON NUMBER
J. LARDIES
Institute FEMTO-ST; DMA; University of Besançon;France
e-mail: [email protected]
Key Words: Vibrations, Stay cables, Cable tension, Scruton number.
Abstract. Identification of modal parameters: eigenfrequencies, damping ratios and mode
shapes from empirical data is of fundamental engineering importance in the dynamical
analysis of mechanical structures and particularly in the analysis of stay cables. These modal
parameters will serve to perform structural health monitoring, damage detection and safety
evaluation. The identification is based on the knowledge of output-only responses, without
using excitation information. We propose in this communication a time domain method based
on a subspace algorithm to extract the modal parameters of vibrating stay cables from output-
only measurements. We can then use these identified parameters for structural health
monitoring by the computation of the Scruton number and the cable tension.
1 INTRODUCTION
Identification of modal parameters from empirical data is very important in
mechanical vibrations and in the dynamical analysis of mechanical structures excited by
external forces. An application constitutes the vibration analysis of stay cables in the civil
engineering domain. The identified modal parameters will serve to perform structural health
monitoring, damage detection and safety evaluation of such structures. The identification is
based on the knowledge of output-only responses, without using excitation information, and is
known as Operational Modal Analysis (OMA), also named as ambient or natural excitation or
output-only modal analysis [1-6]. The principal advantages of OMA are the ease of use and
the fast to conduct since no artificial excitation equipment is required. We propose in this
communication a time domain method, based on a subspace algorithm to extract the modal
parameters of vibrating stay cables from output-only measurements. We can then use these
identified parameters for structural health monitoring.
The subspace identification method is a time domain method and uses the covariance
matrices between output data. The subspace method is based on the observability and
controllability properties of linear systems and we use shifted properties of the controllability
matrix to identify the modal parameters. Unfortunately spurious modes, which are inherent to
the subspace method, appear [6, 7]. To eliminate such spurious poles a criterion using modal
coherence indicators is used. This criterion describes the coherence between each mode of the
state space model and the modes directly inferred from the measured signal. The criterion
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serves as a model quality measure. Modal parameters are then obtained from stability
diagrams. Experimental results of a single tower, double row cable-stayed bridge supported
by 112 stay cables are presented, where ambient vibrations of each stay cable are carried out
using accelerometers. The subspace method is then applied to determine the eigenfrequencies
and damping coefficients of different cables. An estimate of the fundamental frequency fi for
each cable is then obtained. We can then compute the tension force for each cable. For each
natural frequency identified a respective damping coefficient is assigned. Thus, a Scruton
number for each cable and for each mode of vibration is determined [8, 9]. High values of the
Scruton number tend to suppress the oscillation and bring up to the start of instability at high
wind speeds. For example, a Scruton number superior to 10 is sufficient to prevent rain, wind
and traffic induced vibrations. In a continuous monitoring and modal analysis process, the
tension forces and Scruton numbers could be used to assess the health of stay cables in cable-
stayed bridges.
2 VIBRATION ANALYSIS OF A CABLE AND THE SUBSPACE IDENTIFICATION
METHOD
2.1 Vibration analysis of a cable
The free vibration equation of cable motion is given by [10]
EI 4x
t)w(x,4
+ ρ S
2t
t)w(x,2
- P
2x
t)w(x,2
= = f(x,t) (1)
where E is the Young’s modulus, I the moment of inertia of the cable cross-section, ρ the
specific mass, S the cross-sectional area, P the axial load, f(x,t) the external load, w(x,t) the
transversal displacement which is assumed to be small, x the position along the cable and t the
time variable.
The solution of the free vibration equation can be obtained using the method of separation
variables as
w(x,t)=q(t) W(x)
(2)
Assuming q(t) harmonic with natural frequency ω in rad.s-1
and substitution of equation (2)
into equation (1) gives
EI 4x
W4d
- P
2x
W2d
- ρ S ω
2 W = 0 (3)
By assuming the solution W(x) to be
W(x) = C e s x
(4)
in equation (3), the auxiliary equation can be obtained
s4 -
IE
P s
2 -
IE
2ω A ρ = 0 (5)
and the roots of this equation are
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s12, s2
2 =
IE2
P
1/2
IE
2ωAρ
2I2E4
2P
(6)
The solution of W(x) can be expressed as
W(x) = C1 cosh s1 x + C2 sinh s1 x + C3 cos s2 x + C4 sin s2 x (7)
where the constants C1 to C4 are to be determined from the boundary conditions:
W (0) = 0 ; 2xd
W(0)2
d = 0 (8)
W(L) = 0 ; 2xd
W(L)2
d= 0 (9)
Equations (8) require that C1 = 0 and C3 = 0, and so
W(x) = C2 sinh s1 x + C4 sin s2 x (10)
The application of (9) to (10) leads to
sinh s1 L sin s2 L = 0 (11)
Since sinhs1 L > 0 for all values of s1 L 0 , the only roots to this equation are
s2 L = k π , k = 0, 1, 2,….. (12)
Thus equations (12) and (6) give the natural frequencies of vibration:
fk =
1/2
IE2π
2LP2k4k1/2
Sρ
IE
22L
π
(13)
Further, note that the smallest Euler buckling load for a simply supported beam is given by
Pcri=π2
E I / L2 . Thus equation (13) can be written as
fk =
1/2
criP
P2k4k1/2
Sρ
IE
22L
π
(14)
We would like to approximate the desired partial differentiate equation solution w(x,t)
in a separable form as series expansion of time varying coefficients qi(t) and spatially varying
basis functions Wi(x) :
w~ (x,t) =
n'
1i
ii (x)W(t)q (15)
The Galerkin procedure can then be used to specify the equations of motion for the
coefficients qi(t) and for an n’ degrees of freedom system with viscous damping, the motion
equation is
M q + C q + K q = f (16)
where M, C and K are respectively the mass, damping and stiffness matrices. In [11] Barbieri
et al. propose a procedure based on experimental and simulated data to identify damping of
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transmission line cables. The experimental data are collected through accelerometers and the
simulated data are obtained using the finite element method. In this communication, we
identify eigenfrequencies and damping ratios of line cables from output data only using the
subspace method.
2.2 The subspace identification method
The subspace identification method assumes that the dynamic behaviour of a
structure excited by ambient forces can be described by a discrete time stochastic state space
model [1, 2] :
zk+1 = A zk + wk state equation (17)
yk = C zk + vk observation equation (18)
where zk is the unobserved state vector of dimension n=2n’; yk is the (mx1) vector of
observations or measured output vector at discrete time instant k ; wk, vk are white noise terms
representing process noise and measurement noise together with the unknown inputs, it is
assumed that the excitation effect appears in the disturbances wk and vk , since the system
input can not be measured; A is the (nxn) transition matrix describing the dynamics of the
system and C is the (mxn) output or observation matrix, translating the internal state of the
system into observations. The stochastic identification problem deals with the determination
of the transition matrix A using output-only data.
The modal parameters of a vibrating system are obtained by applying the eigenvalue
decomposition of the transition matrix A
A = Λ -1
(19)
where ( )idiag , i=1,2,…,n, is the diagonal matrix containing the complex eigenvalues
and contains the eigenvectors of A as columns. The eigenfrequencies fi and damping ratios
i are obtained from the eigenvalues which are complex conjugate pair:
2
21ln 4 cos
4 2
i ii i i
i i
f Art
(20)
2
2
2
ln
ln 4 cos2
i i
i
i ii i
i i
Ar
(21)
with Δt the sampling period of analyzed signals.
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The mode shapes evaluated at the sensor locations are the columns of the matrix Φ obtained
by multiplying the output matrix C with the matrix of eigenvectors :
Φ = C (22)
Our purpose is to determine the transition matrix A in order to indentify the modal parameters
of the vibrating system. Define the (mfx1) and (mpx1) future and past data vectors as y+
k =
[yT
k, yT
k+1, . . ., yT
k+f-1]T and y
-k = [y
Tk, y
Tk+1, . . ., y
Tk-p+1]
T. The (mfxmp) covariance matrix
between the future and the past is given by [2] :
H = E[y +
k y -kT ] =
1pf1ff
1p32
p21
R.RR
....
R.RR
R.RR
(23)
where E denotes the expectation operator and the superscript T the transpose operation. H is
the block Hankel matrix formed with the (mxm) individual theoretical auto-covariance
matrices Ri = E[yk+i yT
k ] = CAi-1
G, with G=E[xk+1 yT
k].
In practice, the auto-covariance matrices are estimated from N data points and are computed
by Ri = N-1 1-N
1=k
Tki+k
yy ; i = 0,1, . . ., p+f and with these estimated auto-covariance
matrices we form the block Hankel matrix H. In order to identify the transition matrix A two
matrix factorizations of H are employed. The first factorization uses the singular value
decomposition (SVD) of H
H = U Σ VT = U Σ 1/2 Σ 1/2 VT (24)
with UTU and VTV identity matrices and Σ a diagonal matrix of singular values. The second
factorization of the block Hankel matrix H considers its (mfxn) observability and (nxmp)
controllability matrices, O and K, as
H =
GCA.GCAGCA
....
GCA.GCACAG
GCA.CAGCG
2-p+ff1-f
p2
1-p
=
1-fCA
.
CA
C
[G AG. . . Ap-1G] = O K (25)
By identification we obtain the observability matrix
O =
1-fCA
.
CA
C
(26)
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and the controllability matrix
K = [G AG. . . A p-1G] (27)
The two factorizations of the block Hankel matrix are equated to give
H = U Σ 1/2 Σ 1/2 VT = OK (28)
implying O = U Σ1/2
and K = Σ1/2
VT. To determine the transition matrix A we use properties
of the controllability matrix (properties of the observability matrix can also be used). We
introduce the two following matrices:
K = [ AG A2G.....A
p-1G ] and K = [ G AG A
2G.....A
p-2G ] (29)
where K is the matrix obtained by deleting the first block column of K and K is the matrix
obtained by deleting the last block column K. We obtain then:
K = A K or )V (T1/2
Σ = A )V Σ (T1/2
(30)
We use properties of shifting columns operators ’’ ’’ and ’’ ’’ : let Φ and Ψ be two (axb)
and (bxc) matrices, we have the following properties: (Φ Ψ ) =(Φ Ψ ) and
(Φ Ψ ) =(Φ Ψ ), consequently we have
)V Σ ( T1/2
= A )V Σ (T1/2
(31)
The transition matrix is
A = )V Σ ( T1/2
)V Σ (T1/2
+
(32)
where ( )+ represents the pseudo inverse of a matrix, and the eigenvalues of the transition
matrix are given by
)(A = λ [ )V ( T
)V (T
+] (33)
This approach constitutes a subspace modal identification method and in the case of noisy
data a problem of model order determination occurs: when extracting mechanical (or
physical) modes this algorithm can generate spurious modes. For these reasons, the assumed
number of modes, or model order, is incremented over a wide range of values and we plot the
stability diagram. The stability diagram involves tracking the estimates of eigenfrequencies
and damping ratios as a function of model order. As the model order is increased, more and
more modal frequencies and damping ratios are estimated, hopefully, the estimates of the
physical modal parameters stabilize using a criterion based on the modal coherence of
measured and identified modes. Using this criterion we can separate mechanical modes and
spurious modes.
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3 APPLICATION OF THE SUBSPACE IDENTIFICATION METHOD
The subspace identification method is applied to the analysis of stay cables of the Jinma
cable-stayed bridge (Figure 1), that connects Guangzhou and Zhaoqing in Guangdong
Province, China. It is a single tower, double row cable-stayed bridge, supported by 28*4 =112
stay cables. Inputs could evidently not be measured, so only acceleration data from
accelerometers in contact with a cable are available.
Figure 1: View and schematic of the Jimma cable-stayed bridge
For the ambient vibration measurement of each stay cable an accelerometer was mounted
securely to the cables, the sample frequency is 40 Hz and the recording time is 140.8 second,
which results in total 5632 data points. Cables 1, 56, 57 and 112 are the longest and cables
28, 29, 84 and 85 are the shortest. A full description of the test can be found in [8]. Figures 2
and 3 show the stabilization diagrams on eigenfrequencies of cable 1 and cable 25 using the
subspace method and the modal coherence indicator. These diagrams show remarkable stable
eigenfrequencies and from these plots we determine the eigenfrequencies of the two stay
cables.
Figure 2: Ambient time response of cable n°1 and its stabilization diagram on eigenfrequencies
Figure 3: Ambient time response of cable n°25 and its stabilization diagram on eigenfrequencies
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The subspace identification procedure is applied to determine the eigenfrequencies of 112
cables of the bridge and we obtain the Figure 4.
Figure 4: Fundamental frequency of each cable on upstream side and on downstream side
It can be seen that the fundamental frequencies of both bridge sides are almost identical
and the fundamental frequency distribution is symmetric with respect to the single tower. The
fundamental frequencies vary between 0.533 Hz for the longest cable and 2.703 Hz for the
shortest cable. The cable tension can be estimated by the expression T = 4 ρL2 f0
2 , where ρ
is the linear density of the cable (ρ = 66.94 kg/m), L is the length of the cable and f0 the
fundamental frequency. The maximum and minimum cable forces for the Jinma bridge are
then : Tmax = 5052 kN (cable number 57), Tmin = 2490 kN (cable number 84). These cable
forces can be considered as reference tensions and used as indicators in the field of health
monitoring process.
The instability of a cable is assessed by the Scruton number [9] which is defined for each
mode of vibration by , 2
.ic i
a
SD
, where i is the damping ratio for each mode, ρa is the
density of the air and D is the cable diameter. High values of the Scruton number tend to
suppress the oscillation and bring up the start of instability at high wind speeds. Considering
ρa = 1.2 kg/m3 and D = 0.203m, the Scruton number for each mode of cable 25 is presented in
Table 1. In the Scruton number damping is a main factor and most vibration problems in stay
cables are subject to low damping values.
Table 1: Natural frequencies, damping ratios and Scruton numbers for cable 25
Modes 1 2 3 4 5 6
fk (Hz) 1.960 3.938 5.915 7.893 9.856 11.884
k (%) 0.314 0.046 0.178 0.078 0.081 0.094
Sck 4.250 0.622 2.410 1.056 1.096 1.271
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4 CONCLUSION
Cable vibrations induced by wind, rain and traffic are observed in a number of cable-
stayed bridges and degradation of these cables may have catastrophic effects. The
determination of tension and Scruton number of these cables from output only measurements
can be used as structural health monitoring indicators. Operational modal analysis applied to
the dynamic data of stay cables provides useful information to determine the current condition
of stay cables accurately.
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